Cycles frustrating fractal formation in an AB2 step growth polymerization

Abstract

The step growth of a flexible AB ₂ monomer, normally considered to produce dendrimers or fractal molecules, as they are themselves self-similar to their branches, is frustrated by the formation of one cycle within each molecule, according to a Monte Carlo lattice study of the evolution of the topological trees that are embedded in three-dimensional space; the number of cycles of m residues is well fitted by the relationship R _m = K ₀ p _a ^m m ^{- e} , p _a being the extent of reaction of the A groups; at the end of a polymerization of N ₀ monomers the total number of cycles, and of molecules, is given by the product of K ₀ and the Euler–Riemann function ξ(e) so there is a simple relationship between these quantities and the mean number average degree of polymerization at infinite time: N ₀ = K ₀ <x> _n,∞ ξ(e).